C^0 lower semi-continuity conjecture for entropy of Reeb flows

Let $(Y,\xi)$ be a closed co-oriented contact $3$-manifold, and let $\mathcal{R}(Y,\xi)$ denote the space of Reeb flows on $(Y,\xi)$. Consider the entropy functional

$$h_{\rm top}:\mathcal{R}(Y,\xi)\to[0,\infty).$$

Reeb-flow entropy stability conjecture. The functional $h_{\rm top}$ is lower semi-continuous with respect to $d_{C^0}$.

This conjecture proposes $C^0$-stability of topological entropy for Reeb flows in dimension $3$, extending the paper's study of entropy stability in symplectic and contact settings. Its resolution is not specified in the supplied text.